Chapter 5 | Integration
587
Write the integral in terms of u , but pull the 1 2
outside the integration symbol:
⌠ ⌡ z
1/2
⎛ ⎝ z 2 −5
⎞ ⎠
2 ∫
u 1/2 du .
dz = 1
Integrate the expression in u :
⎛ ⎝ 1 2 ⎛ ⎝ 1 2
⎞ ⎠ u
3/2 3 2
1 2 ∫
u 1/2 du =
+ C
⎛ ⎝ 2 3
⎞ ⎠ u 3/2 + C
⎞ ⎠
=
u 3/2 + C
= 1 3 = 1 3
3/2
⎛ ⎝ z 2 −5
⎞ ⎠
+ C .
Use substitution to find ⌠ ⌡ x 2 ⎛
9
5.26
⎞ ⎠
⎝ x 3 +5
dx .
Example 5.32 Using Substitution with Integrals of Trigonometric Functions Use substitution to evaluate the integral ⌠ ⌡ sin t cos 3 t dt .
Solution We know the derivative of cos t is −sin t , so we set u =cos t . Then du =−sin tdt . Substituting into the integral, we have
⌠ ⌡
dt =− ⌠ ⌡
du u 3
sin t cos 3 t
.
Evaluating the integral, we get
− ⌠ ⌡
du u 3
=− ∫ u −3 du
⎛ ⎝ − 1 2
⎞ ⎠ u −2 + C .
=−
Putting the answer back in terms of t , we get ⌠ ⌡
sin t cos 3 t
dt = 1
+ C
2 u 2
= 1
+ C .
2cos 2 t
Made with FlippingBook - professional solution for displaying marketing and sales documents online